Runge-Kutta Procedures Research Articles

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.

Unsteady radiative magnetohydromagnetic flow and entropy generation of maxwell nanofluid in a porous medium with arrhenius chemical kinetic

Parametric sensitivity of unsteady Maxwell magnetohydromagnetic nanofluid and second thermodynamic law analysis under Arrhenius kinetic is investigated in the presence of viscous heating and radiation. The porous flow channel is subjected to tension with material properties that vary with time without deformation. In the absence of fluid charge polarization, the conducting liquid is influenced by the sheet stretching velocity. The flow coupled derivatives model is reduced to dimensionless form by relevant transformation variables. These are computational solved by shooting numerical technique together with Fehlberg Runge-Kutta procedures. The essential characteristics of the flow and thermodynamic irreversibility are determined. The results are quantitatively and qualitatively compared with other studies and are established to agree well. The graphical results revealed that Lewis number increases the molecular species concentration and the thermodynamic stability for reversibility can be enhanced by the augmentation of magnetic field, thermophoresis, and radiation. Therefore, for thermal and chemical reaction systems, increasing heat propagation should be managed to keep the system from blowing up.

Properties-preserving high order numerical methods for a kinetic eikonal equation

For the BGK (Bhatnagar–Gross–Krook) equation in the large scale hyperbolic limit, the density of particles can be transformed as the Hopf–Cole transformation, where the phase function converges uniformly to the viscosity solution of an effective Hamilton–Jacobi equation, referred to as the kinetic eikonal equation. In this work, we present efficient high order finite difference methods for numerically solving the kinetic eikonal equation. The methods are based on monotone schemes such as the Godunov scheme. High order weighted essentially non-oscillatory techniques and Runge–Kutta procedures are used to obtain high order accuracy in both space and time. The effective Hamiltonian is determined implicitly by a nonlinear equation given as integrals with respect to the velocity variable. Newton's method is applied to solve the nonlinear equation, where integrals with respect to the velocity variable are evaluated either by a Gauss quadrature formula or as expansions with respect to moments of the Maxwellian. The methods are designed such that several key properties such as the positivity of the viscosity solution and the positivity of the effective Hamiltonian are preserved. Numerical experiments are presented to demonstrate the effectiveness of the methods.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

Laplace based approximate posterior inference for differential equation models

Ordinary differential equations are arguably the most popular and useful mathematical tool for describing physical and biological processes in the real world. Often, these physical and biological processes are observed with errors, in which case the most natural way to model such data is via regression where the mean function is defined by an ordinary differential equation believed to provide an understanding of the underlying process. These regression based dynamical models are called differential equation models. Parameter inference from differential equation models poses computational challenges mainly due to the fact that analytic solutions to most differential equations are not available. In this paper, we propose an approximation method for obtaining the posterior distribution of parameters in differential equation models. The approximation is done in two steps. In the first step, the solution of a differential equation is approximated by the general one-step method which is a class of numerical methods for ordinary differential equations including the Euler and the Runge-Kutta procedures; in the second step, nuisance parameters are marginalized using Laplace approximation. The proposed Laplace approximated posterior gives a computationally fast alternative to the full Bayesian computational scheme (such as Markov Chain Monte Carlo) and produces more accurate and stable estimators than the popular smoothing methods (called collocation methods) based on frequentist procedures. For a theoretical support of the proposed method, we prove that the Laplace approximated posterior converges to the actual posterior under certain conditions and analyze the relation between the order of numerical error and its Laplace approximation. The proposed method is tested on simulated data sets and compared with the other existing methods.

Quaternion-based dynamics of geometrically nonlinear spatial beams using the Runge–Kutta method

The rotational quaternions are the unique four-dimensional representation of rotations in three-dimensional Euclidean space. In the present paper on the dynamics of non-linear spatial beams, they are used as the basic rotational parameters in formulating the finite-element approach of geometrically exact beam-like structures. The classical concept of parametrizing the rotation matrix by the rotational vector is completely abandoned so that the only rotational parameters are the rotational quaternions representing both rotations and rotational strains in the beam. Because the quaternions are the elements of a four-dimensional linear space, their use is an advantage compared to the elements of the special orthogonal group SO(3). This makes possible, e.g. to interpolate the rotational quaternions in a standard additive way and to apply standard Runge–Kutta time integration methods. The present finite-element realization of such a quaternion-based beam theory introduces some further ideas including the use of collocation, the weak form of the consistency equations and the symmetrically integrated boundary conditions which enhances the accuracy or efficiency of the solution and avoids the differential–algebraic type of the discretized governing equations often met in classical formulations. The main advantage of the proposed dynamics formulation is probably its simple implementation into the standard Runge–Kutta procedures. Thus several benefits of the Runge–Kutta methods such as the absence of the analytical linearization of the discrete governing equations, the local error control and the adaptive time steps are automatically incorporated in the present procedure.

Stepsize restrictions for total-variation-boundedness in general Runge–Kutta procedures

In the literature, on the numerical solution of nonlinear time dependent partial differential equations, much attention has been paid to numerical processes which have the favourable property of being total variation bounded (TVB). A popular approach to guaranteeing the TVB property consists in demanding that the process has the stronger property of being total variation diminishing (TVD). For Runge–Kutta methods—applied to semi-discrete approximations of partial differential equations—conditions on the time step were established which guarantee the TVD property; see, e.g., [J. Comput. Phys. 77 (1988) 439; Math. Comp. 67 (1998) 73; SIAM Rev. 43 (2001) 89; SIAM J. Numer. Anal. (2002), in press; Higueras, Tech. Report, Universidad Pública de Navarra, 2002; SIAM J. Numer. Anal. 40 (2002) 469]. These conditions were derived under the assumption that the simple explicit Euler time stepping process is TVD. However, for various important semi-discrete approximations, the Euler process is TVB but not TVD—see, e.g., [Math. Comp. 49 (1987) 105; Math. Comp. 52 (1989) 411]. Accordingly, the above stepsize conditions for Runge–Kutta methods are not directly relevant to such approximations, and there is a need for stepsize restrictions with a wider range of applications. In this paper, we propose a general theory yielding stepsize restrictions which cover a larger class of semi-discrete approximations than covered thus far in the literature. In particular, our theory gives stepsize restrictions, for general Runge–Kutta methods, which guarantee total-variation-boundedness in situations where the Euler process is TVB but not TVD.

Solving stiff differential equations in the simulation of physical systems

Computer simulation of dynamic systems often requires the solution of a set of stiff ordinary differential equations. The solution involves the eigenvalues of the Jacobian matrix. The greater the spread in eigenvalues, the more time-consuming the solutions become if one uses standard numerical techniques. In many cases, the differential equations are so stiff that accurate numerical solutions are unecon omic. In this paper we propose new techniques for solving stiff differential equations. These algo rithms are based on implicit Runge-Kutta procedures with complete error estimates. Experiments show that the new techniques are superior to Gear's method for solving mathematical models for real sys tems in chemical engineering, biology, and laser kinetics.

Stability analysis of certain Runge-Kutta procedures for Volterra integral equations

Stability analysis of certain Runge-Kutta procedures for Volterra integral equations

On a Class of Implicit Runge-Kutta Procedures

On a Class of Implicit Runge-Kutta Procedures

A Note on the Computational Aspects of a Class of Implicit Runge-Kutta Procedures

A Note on the Computational Aspects of a Class of Implicit Runge-Kutta Procedures

Semi-Implicit Runge-Kutta Procedures with Error Estimates for the Numerical Integration of Stiff Systems of Ordinary Differential Equations

A -stable, semi-implicit Runge-Kutta procedures requiring at most one Jacobian evaluation per time step are developed for the approximate numerical integration of stiff systems of ordinary differential equations. A simple procedure for estimating the local truncation error is described and, with the help of this estimate, efficient integration procedures are derived. The algorithms are illustrated by direct application to a particular example.

Comparing error estimators for Runge-Kutta methods

A way is proposed to compare local error estimators. This is applied to the major estimators for fourth-order Runge-Kutta procedures. An estimator which leads to a production code 18

A Generalization of a Theorem of Carr on Error Bounds for Rung-Kutta Procedures

In [1] Carr established propagation error bounds for a particular Runge-Kutta (RK) procedure, and suggested that similar bounds could be established for other RK procedures obtained by choosing the parameters differently.